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Nonlinear Dynamics, 2013

Homework 1

1. Consider the linear systems described by ( ) ( )t tx Ax , with the system matrix given by:

(i) 16 3

2 1

A ; (ii) 22 1

4 3

A ; (iii) 30 1

1 0

A ; (iv) 41 1

1 1

A ;

(v) 51 2

0 1

A .

a. Find the explicit solution for some initial condition 20 x .

b. Classify the stability of the origin. c. Draw the phase portrait.

2. Consider the nonlinear systems

(i) 1 1 1 2

22 1 2

2

2

x x x x

x x x

;(ii)

2 31 1 2 1 2 2

3 2 22 1 2 1 2

x x x x x x

x x x x x

.

a. Find all the equilibrium points. b. Simulate a number of trajectories of the system with initial conditions fixed in the

vicinity of the equilibrium points.

c. Draw the phase portrait. d. Determine the type of stability for each isolated equilibrium point.

3. Consider the nonlinear system

2 2 1 21 1 2 1 1 2

2 21 2

22 2 1

2 1 2 2 1 22 21 2

( )

( )

x xx x x x x x

x x

xx x x x x x

x x

a. Show that (1,0) is an equilibrium point. b. Simulate a number of trajectories of the system with initial conditions fixed in the

vicinity of the equilibrium point.

c. Draw the phase portrait. d. Determine the type of stability for the given equilibrium point.

4. Consider the nonlinear system

31 2 1

32 2 2 1

4

3

x x x

x x x x

a. Find all the equilibrium points. b. Using linearization, classify the stability of the fixed points.

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c. Show that the line 1 2x x is an invariant set in the sense that any trajectory that starts

on this line stays on the line forever.

d. Show that 1 2lim | ( ) ( ) | 0t x t x t for all state trajectories of the system (Hint: write

the differential equation satisfied by 1 2x x ).

e. Draw the phase portrait on the domain [ 20,20] [ 20,20] and plot a number of

trajectories of this system with initial conditions fixed in on the boundary of the

considered domain.

f. Comment on the behavior of this system.

5. Consider the system

1 2 3

2 1 3 2

23 3 3

1

(1 )

x x x

x x x x

x x x

a. Show that the system has a unique equilibrium point. b. Using linearization, show that the equilibrium point is asymptotically stable. Is it

globally asymptotically stable?

Homework 2

6. Using a quadratic Lyapunov function show that the origin is a locally asymptotically stable

equilibrium point for the system

21 1 1 2

2 1

x x x x

x x

corrected

21 1 1 2

2 1

x x x x

x x

7. Consider the system

1 1

3 22 1 2 2 1 2 1 2( 1) ( 1 )

x x

x x x x x x x x

a. Show that the system has a unique equilibrium point. b. Using linearization, show that the equilibrium point is asymptotically stable.

c. Show that the set 2 1 2| 2x x x is positively invariant with respect to the system.

8. Consider the system

31 1 1 2

2 1 23

x x x x

x x x

a. Find all the equilibrium points. b. Using linearization, study the stability of each equilibrium point.

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c. Using quadratic Lyapunov functions, estimate the region of attraction of each asymptotically stable equilibrium point.

d. Draw the phase portrait of the system and show on it the exact regions of attraction as well as your estimates.

9. Consider the controlled nonlinear system represented in the figure below with (i) 2

1( )s

P s ,

or, equivalently, x u ; (ii) 1( 1)

( )s s

P s

, or, equivalently, x x u .

a. Give the state space description of the closed loop system, considering the states

1x x and 2x x .

b. Find all the equilibrium points. c. Using linearization, study the stability of each equilibrium point. d. Draw the phase portrait of the nonlinear system. e. Verify if the origin is stable by means of the Lyapunov stability theorem using the

Lyapunov function candidate 4 21 11 2 1 24 2( , )x x x x .

u x x